COLLECTIVE COHN-VOSSEN PROBLEM,
PARABOLIC LOCALIZATION PRINCIPLE ANDPERIOD CONDITION OF ALGEBRAIC MINIMAL SURFACES

Ryoichi Kobayashi (joint with Reiko Miyaoka)
Abstract. According to the interpretation proposed in [KKM] of Osserman’s theory of algebraic minimal
surfaces [O1,2], this theory studies the value distribution of the Gauss map of algebraic minimal surfaces by
estimating the basic ratio R, which is the ratio of the Fubini-Study area against the hyperbolic area, of thealgebraic minimal surface under question, by using the Riemann-Roch (or Gauss-Bonnet) formula under thepresence of the period condition. In this paper, we further develop the approach of [KKR] :(i) We propose a “partition function” Z1−r involving a parameter 1 − r (0 < r < 1 being the radius of
the disk |z| < r) which formally yields the basic ratio R if we take the limit 1 − r → 0 before performing
the (potentially inﬁnite) sum which counts states with certain weights. We reveal what kind of mathematics
emerges if we interchange the order of taking the (potentially inﬁnite) sum and taking the limit 1 − r → 0
into the standard order in the semi-classical limit. We show that this change of limits, i.e., the study of
the asymptotic behavior of the partition function Z1−r when 1 − r → 0 (r → 1) reduces to the Nevanlinna
Theoretical study of the Weierstrass data.
(ii) We propose a version of the Nevanlinna theory coupled with the π1 (M )-action on the unit disk
D = {|z| < 1} where π1 (M ) is the fundamental group of the algebraic minimal surface under question.
This theory, which we call the Nevanlinna-Galois theory, ﬁts to the purpose of investigating the asymptotic
behavior of the partition function Z1−r . We reveal the prominent role of the action of π1 (M ) (a free Fuchsiangroup) on D in the classical minimal surface theory whose eﬀect on the Nevanlinna-Galois theory culminatesin the “collective Cohn-Vossen inequality”. As an application, we establish an eﬀective version of the Lemma
on Logarithmic Derivative (LLD) valid for the lifted Gauss map of algebraic minimal surfaces and more1
whengenerally for meromorphic functions on D whose height transform has comparable growth as log 1−r
r → 1.
(iii) We propose a Nevanlinna theory interpretation of the period condition of algebraic minimal surfaces.
More precisely, we propose a canonical construction, from the Weierstrass data, of a pair (eH , D) of aholomorphic function eH on D and a divisor D (potentially of inﬁnite degree) on P1 so that eH maximallyapproximates D and the pair (eH , D) fully encodes the period condition. We show that we can apply oureﬀective LLD to the pair (eH , D). Combining these two results, we establish Nevanlinna’s Second Main
Theorem for the Gauss map and give the optimal upper bound for the totally ramiﬁed value number of thelifted Gauss map of algebraic minimal surfaces. In particular, we prove that the Gauss map of any algebraic
minimal surface can omit at most two values.

§1. Introduction.
This note is an extended version of my lecture delivered at 2019 Oka Symposium. My lecture was basedon the preprints [KM1,2]. These papers are very much involved. Therefore, the purpose of this note
is to introduce main ideas in [KM1,2] with geometric intuitions behind results in [KM1,2] rather thanproofs. In the course of preparing this note, I discovered the reason why the Cohn-Vossen ratio R (fordeﬁnition, see below) being 2 is so special in the theory of algebraic minimal surfaces. In this note, I
will put emphasis on this discovery.
We study the period condition of algebraic minimal surfaces from the view point of the action of afree Fuchsian group action on D and Nevanlinna theory. An open Riemann surface M on which theWeierstrass data (g, ω) is deﬁned is called the basic domain. We write (M, (g, ω)) for an algebraic
minimal surface, i.e., a complete minimal surface in R3 with ﬁnite total curvature, where M is an open
Riemann surface, (g, ω) the Weierstrass data, i.e., g : M → P1 the Gauss map and ω a holomorphic
1-form on M satisfying the regularity condition and the period condition. The algebraic minimal surface
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RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)

M is immersed in R3 by the Ennneper-Weierstrass representation formula
Z z
1 − g21 < i(1 + g 2 )  ω ∈ R3 .
M 3 z 7−→z0 2
2g
The regularity condition means that the induced metric ds2 = 14 (1 + |g|2 )2 )|ω|2 is positive, i.e., g has apole of order m at P ∈ M if and only if ω has zero of order 2m. The period condition means that the
Enneper-Weierstrass representation is single-valued, i.e., we haveZ
1 − g21 i(1 + g 2 )  ω ∈ i R3 , ∀γ ∈ H1 (M, Z) .
2 γ
2g
We denote ωFS (resp. ωhyp ) for the Fubini-Study metric on P1 (resp. the Poincaré metric on D)
normalized so that the absolute value of the Gaussian curvature is 1. In [KKM], the authors haveshown that the Cohn-Vossen ratioR− M g ∗ ωFS
R
R=
ω
M hyp
plays the essential role in Osserman’s theory of algebraic minimal surfaces [O1,2]. Namely, Osserman’s
main results are expressed in terms of the Cohn-Vossen ratio by
R>1( Riemann-Roch + period condition )and2( Riemann-Hurwitz )R
for any algebraic minimal surface. Here νg is the totally ramiﬁed value number of the Gauss map (cf.
[KKM]). In particular we have νg < 4 and therefore the Gauss map can omit at most three values. It isa folklore conjecture that the Gauss map of an algebraic minimal surface can omit at most two values.
This conjecture is called Osserman’s problem. If R > 2 then νg < 3 holds. In particular, the Gauss
map g can omit at most two values. Therefore, to settle Osserman’s problem, it suﬃces to show thatνg < 3 holds under the assumption R ≤ 2. In [KM2, Part II], the authors obtained the Second Main
Theorem for the Gauss map of an algebraic minimal surface (M, (g, ω)) under the assumption R ≤ 2.
The estimate νg < 3 follows from the Second Main Theorem.
νg ≤ 2 +The purpose of the present note is to prove the Second Main Theorem for the (lifted) Gauss mapof an algebraic minimal surface without assuming anything on the Cohn-Vossen ratio. Our strategy inthis note is essentially the same as in [KM2, Part II]. We will follow the argument in [KM2, Part II].
At every place where we use inequalities proved in [KM2, Part II] under the assumption R ≤ 2, we
replace them with more general (but weaker) ones which hold without assuming anything on R. The
main point is Theorem 2.8.3.1 which states that Ng,∞ (r) (and therefore mg,∞ (r) also) behaves in a
special way relative to Tg (r), where ∞ (resp. 0) 6∈ {g(P ) | P ∈ M \ M or dg(P ) = 0} is the repelling(resp. absorbing) ﬁxed point of the hyperbolic translation T and the metric of P1 on which Tg (r) is
modiﬁedmodiﬁeddeﬁned is the pillow-case metric ωFS, where T and ωFS
appeared in the proof of Metrized
Riemann-Hurwitz Theorem [Part I, Theorem 4.2] (see 2.4 of the present note).
Theorem 1.1 (Theorem 2.8.3.1) (parabolic localization in terms of group theoretic approximation). Suppose that (M, (g, ω)) be a pseudo-algebraic minimal surface. For ∞ 6∈ {g(P ) | P ∈M \ M or dg(P ) = 0} which appears in the proof of Metrized Riemann-Hurwitz Theorem [Part I, Theorem 4.2], we have
Ng,∞ (r) ∼ O(n−1th ) Tg (r) ,mg,∞ (r) ∼ (1 − O(n−1th )) Tg (r) .
modiﬁedHere the metric on P1 with which Tg (r) is deﬁned is the pillow-case metric ωFS
in Metrizd RiemannHurwitz Theorem [Part I, Theorem 4.2].

COLLECTIVE COHN-VOSSEN PROBLEM

3

Here nth is the threshold of the Euclidean distortion introduced in [KM1] (for deﬁnition, see 2.3 ofthe present note), which can be taken arbitrarily large positive number. Theorem 1.1 is a Nevanlinna
Theoretic interpretation of the parabolic localization principle introduced in [KM1,2] (see 2.3 of thepresent note). The “parabolic localization” was introduced in [KM1] which is a sort of localization
phenomenon arising from the action of the free Fuchsian group π1 (M ) on D (for a brief discussion, see
2.3 of the present note). For the meaning of the “group theoretic approximation”, see 2.8.3 of thepresent note.
The Main Theorem of the present note (in fact, the same as that of [KM2]) is the following :Theorem 1.2 (Theorem 2.10.6) (Second Main Theorem for the Gauss map of an algebraic
minimal surface). Let D = {a1 , . . . , aq } be a set of distinct points of P1 . The lifted Gauss map
g : D → P1 of an algebraic minimal surface (M, (g, ω)) satisﬁes the Second Main Theorem
mg,D (r) + Ng,Ram (r) ≤ {4(e − 2) + ε)} Tg (r) ,where ε is any small positive number.
In particular, we have νg ≤ 4(e − 2) ≤ 2.88 and therefore the Gauss map g can omit at most twovalues.
§2. Proof of Main Theorem.
2.1. Partition Function.
Let M be an open Riemann surface and (g, ω) a Weierstrass data on M , where g : M → P1 is the (stereographically projected) Gauss map and ω is a holomorphic 1-form satisfying the regularity condition,i.e., g has a pole of order m at P ∈ M if and only if ω has zero of order 2m at the same point. The
Enneper-Weierstrass representation
Z z
1 − g21 < i(1 + g 2 )  ω ∈ R3
M 3 z 7−→ X(z) :=z0 2
2g
deﬁnes a holomorphic map from the universal covering surface of M into R3 . We say that (M, (g, ω)) isa pseudo-algebraic minimal surface if M is a ﬁnite Riemann surface, i.e., M = M \ {P1 , . . . , Pn(M ) }where M is a compact Riemann surface, and the Weierstrass data (g, ω) extends meromorphically acrossthe punctured points {P1 , . . . , Pn(M ) }. A pseudo-algebraic minimal surface is an algebraic minimal
surface if in addition the period condition is satisﬁed :Z
1 − g21 i(1 + g 2 )  ω ∈ i R3 , ∀γ ∈ H1 (M, Z) ,2 γ
2gi.e., the Enneper-Weierstrass representation X(z) is single-valued (by Huber [Hu] and Osserman [O1,2],
this deﬁnition of algebraic minimal surface coincides with the classical one). Throughout this paper we
f = D, D = {z ∈ C | |z| < 1} is the unit disk in C.
assume that the universal covering surface M
The ﬁrst step toward the Second Main Theorem for the lifted Gauss map g : D → P1 is to replace
the Cohn-Vossen ratio R by the partition function Z deﬁned byZ(r) =

Xα∈π1 (M )AreaFS (Fα ∩ D(r)).
Areahyp (D(r))

As soon as we ﬁx a reference fundamental domain F , we have a one to one correspondence between
α ∈ π1 (M ) and a fundamental domain Fα = α(F ). We then count the number of elements α of π1 (M )with the Cohn-Vossen type weight
AreaFS (Fα ∩ D(r)), D(r) = {z ∈ D | |z| < r}
Areahyp (D(r))

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RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)

to deﬁne the partition function Z(r). In particular, for a ﬁxed 0 < r < 1, the sum in Z(r) is ﬁnite.
The idea behind the replacement R 7→ Z(r) is to embed Osserman’s problem into the realm of the free
Fuchsian group (π1 (M ))-action and Nevanlinna Theory on D by lifting the Weierstrass data (g, ω) tothe universal covering D without looking at individual algebraic minimal surface M = π1 (M )\D with
unlifted Weierstrass data (g, ω). The number 1 − r plays the role of the Planck constant. The study ofthe semi-classical limit limr→1 Z(r) reduces to the comparison of following two Nevanlinna Theoretic1
1functions Tg (r) and Thyp (r), i.e., the height transform of 4π
ωFS and 4π
ωhyp , in the presence of theπ1 (M )-action on D. Here,√dz ∧ dz
ωFS = 2 −1
on P1 ,
(1 + |z|2 )2√dz ∧ dzon D1 .
ωhyp = 2 −1
(1 − |z|2 )2
are the Fubini-Study metric and the Poinbcaré metric normalized so that the absolute value of theGaussian curvature is 1. Explicitly Tg (r) and Thyp (r) are deﬁned as
Z r
Z
dt
1 ∗
Tg (r) =g ωFS ,
t
4π
0
D(t)
Z r
Z
1
dt
1
asymptotically as r→1 1
Thyp (r) =
ωhyp
=
log
.
t
4π
2
1
−
r
0
D(t)
2.2. Period Condition.
Every minimal surface is locally represented by the Enneper-Weierstrass representation formula. The
freedom of rotating a minimal surface in R3 corresponds to the freedom in the choice of a unit tangent
vector from the tangent bundle of


1
i
3
2
2
2
2
2
S := a (1 − g ) + b (1 + g ) + cg (a, b, c) ∈ R , a + b + c = 1 ,22
where S is the unit sphere of the 3-dimensional R-vector space V spanned by the ON basis { 21 (1 −g 2 ), 2i (1 + g 2 ), g}. Therefore, instead of working with three components in the Enneper-Weierstrass
representation formula, we choose to work with a random polynomial p(g) in g chosen from S. Givenp(g) ∈ S we deﬁneZz

H(z) :=

p(g)ω ,z0

which deﬁnes a holomorphic function on D by the regularity condition. If z0 is a reference point in D
and α ∈ π1 (M ), then the period condition implies that there exists a γ ∈ H1 (M, Z) such thatZ
H(αz) − H(z) =p(g)ω ∈ i R3 .
γ

Therefore|eH (z)| = |eH (αz)|∀z ∈ D and α ∈ π1 (M ) .

Thus we have
Lemma 2.2.1. The period condition is equivalent to saying that |eH | is invariant under the action ofπ1 (M ).
2.3. Parabolic Localization.
In 2.3, we do not need the period condition. Hence we work on pseudo-algebraic minimal surfaces.
In order to study the semi-classical limit limr→1 Z(r) in terms of the Nevanlinna Theory, we need toanalyze the deviation from the action of π1 (M ) to D from the Euclidean similarity. This is because theconcentric disks |z| < r and their boundaries |z| = r are involved in every Nevanlinna theoretic function

COLLECTIVE COHN-VOSSEN PROBLEM

5

so that every such function is a function in r. As D and H = {z ∈ C | =(z) > 0} are conformallyequivalent via the Cayley transform H 3 w 7→ z = w−iw+i ∈ D, we may work on H and translate obtained
results into D.
In order to analyze the Euclidean distortion of fundamental domains, we considerxT (x) =
x+1
which is a parabolic translation with a unique ﬁxed point 0 ∈ ∂H (∂H contains ∞). By iteration wehavex
T n (x) =.
nx + 1
We call the iteration of a parabolic translation a parabolic sequence. Let I = [a, b] ⊂ R+ = {x ∈ R | x >0} be a closed interval in R+ = {x ∈ R | x > 0}. We imagine a fundamental domain F having 0 andend points {a, b} of I as three vertices and all other vertices locate in the interior (a, b) of I. We set
In := T n (I) and call it the cluster part. A simple computation shows thatdist euc (0, In ) = O(n−1 )anddiam euc (In ) = O(n−2 ) .

Visually this is interpreted as follows. Look at the image of ∂D(r) = {z ∈ D | |z| = r} in M for r < 1
close to 1. Then as soon as the image path γ approaches to a punctured point P (corresponding toa parabolic ﬁxed point), the image curve γ rotates many times around P and stays longer. This is
the localization induced from the parabolic sequence. We call this localization phenomenon as theparabolic localization principle. One of the basic fact is the non-occurrence of the multi-cluster
part.
Lemma 2.3.1 [KM1, Part I, Lemma 3.4.1.4] (non-existence of multi-cluster part). There is
at most one cluster part in every fundamental domain.
Lemma 2.3.1 implies that the following deﬁnition makes sense. We call the ratio
dist euc (0, In )diam euc (In )the Euclidean distortion. We ﬁx a large number nth and say that a fundamental domain is Euclidean
distorted if the Euclidean distortion is larger than nth (nth is the threshold of Euclidean distortion).
The parabolic localization means the distortion in the Euclidean sense induced by the iteration of aparabolic translation and the truncation of the Euclidean distorted fundamental domains by D(r), r
being comparable to the Euclidean height of the cluster part, has the eﬀect of isolating the neighborhoodof (more precisely, the annular domain around) the parabolic ﬁxed point under question. This is theorigin of the name “parabolic localization”. In order to get useful information from the parabolic
localization principle, knowing the contribution from parabolic sequences in any measure theoretic
quantity associated to D(r) as well as the asymptotic behavior of the number N1 (resp. N2 ) of the
Euclidean distorted (resp. non-distorted) fundamental domains is essential1 . For this, we have thefollowing basic results :Lemma 2.3.2 [KM1, Part I, Lemma 3.4.3.7] (dominating parabolic sequences covering
D(r)). Set
2 log nthα(r) := 1 −.
1
log 1−rThen the iteration images of fundamental domains of parabolic translations contained in the minimal
covering of D(r) is dominated by those parabolic sequences starting at fundamental domains of Euclidean
diameter (1 − r)α(r) .
Lemma 2.3.2 indicates that we can count any measure theoretic quantity associated to D(r) in thefollowing way.
1 We count the number N (resp. N ) of Euclidean distorted (resp. non-distorted) fundamental domains in the12minimal covering by fundamental domains of D(r) and therefore Ni should be interpreted as Ni (r) (i = 1, 2).

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RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)

Lemma 2.3.3 [KM1, Part I, Lemma 3.4.3.8] (counting rule of quantities associated toD(r)). In order to count any measure theoretic quantity associated to D(r), we ﬁrst classify all parabolic sequences according to the Euclidean diameter of the initial Euclidean non-distorted fundamental
domain. Then we count the contribution from individual parabolic sequence starting at a Euclidean nondistorted fundamental domain of Euclidean diameter O(1−t). Then we multiply the number O((1−t)−1 )of parabolic sequences starting at a Euclidean non distorted fundamental domain with Euclidean radiusO(1 − t) and integrate the result against the measuredµ(t) :=dt1−tfrom, say, t = 2−1 to t = 1 − (1 − r)α(r) .
For instance, we count the quantity in question in the case that the contribution from individual
parabolic sequence is 1. In this case we have
Z 1−(1−r)α(r)121dµ(t) =1−t

Z 1−(1−r)α(r)
1
2

1
dt
=
.
(1 − t)2(1 − r)α(r)This counts the number of the fundamental domains of Euclidean radius O(1 − r)α(r) .
Lemma 2.3.4 [KM1, Part I, Lemma 3.4.3.9-10] (comparison of numbers of Euclidean distorted / non-distorted fundamental domains covering D(r)). Fix any large number nth as thethreshold of the Euclidean distortion. Then we have1
N1
= O(nth (log nth ) 2 )N2

as r → 1. This means that the contribution from Euclidean distorted fundamental domains asymptotically dominates as r → 1 the signiﬁcant portion of the minimal covering of D(r) by truncatedfundamental domains (the intersection of fundamental domains and D(r)).
In particular, the ratio ofR
the contribution from parabolic sequences against the Fubini-Study area D(r) g ∗ ωFS of D(r) is bounded
below by 1 − ε for any ε > 0 uniformly when r → 1.
As for the uniformity of the cluster part, we have
Lemma 2.3.5 [KM1, Part I, Lemma 3.4.1.5] (uniformity of cluster part). The variation of
the distribution of ﬁnitely many parabolic ﬁxed points in the cluster part is uniform.
Lemma 2.3.5 implies that hyperbolic translations in π1 (M ) do not contribute to the formation ofcluster parts [KM1, Part I, Lemma 3.4.2.1]. Using Lemma 2.3.5, we can introduce a dynamical systemon the boundary S 1 = ∂D deﬁned by the evolution of orbits of fundamental domains and the conceptof the scale. The existence of such a dynamical system is the background geometry developed in later
sections.
Deﬁnition 2.3.6 [KM1, Part I, Deﬁnition 3.4.2.2](A dynamical system on S1 induced fromthe action of the free Fuchsian group π(M ) and the associated scale on S1 ). We ﬁx a reference
fundamental domain F0 wrt. the action of π1 (M ) where M is the basic domain of a pseudo-algebraic
minimal surface. Then F0 is realized as an ideal geodesic (4G + 2(n(M ) − 1))-gon with vertices on
S1 = ∂D. We can then introduce the dynamical system on S 1 = ∂D arising from the vertices of fundamental domains obtained by applying words of π1 (M ), i.e., the evolving orbit of vertices of fundamental
domains. The uniformity of the cluster part enables us to introduce the concept of the scale on S1
by saying that this dynamical system is in the scale ρ > 0 if the mesh of the distributed vertices iscomparable to ρ.
2.4. Metrized Riemann-Hurwitz Theorem.
We assume that the Gauss map g : M → P1 has no poles at the punctured points and all poles are
simple, i.e., 0, ∞ 6∈ {g(P ) | P ∈ M \ M and dg(P ) 6= 0}. The parabolic localization principle truncatedCOLLECTIVE COHN-VOSSEN PROBLEM

7

at D(r) isolates neighborhoods of the punctured points. Therefore, we need to establish a tool whichallows us to get useful information from local computation around punctured points. The tool weneed is the Metrized Riemann-Hurwitz Theorem. To formulate it, we need to introduce a certain celldecomposition of the basic domain of a pseudo-algebraic minimal surface. This is outlined as follows.
The 1-form part ω = hdz of the Weierstrass data should be understood as a holomorphic map sending∂∂
the coordinate vector ∂z
to the tangent vector h(z) ∂w
in Tg(z) P1 . So, we can compare dg and h as
1linear maps Tz M to Tg(z) P at each z ∈ M .
Let T be a hyperbolic translation of P1 having ∞ (resp. 0) as a repelling (resp. absorbing) ﬁxed
point. We replace the stereo-graphically projected Gauss map g : M → P1 with the compositiongT

T (g) : M → P1 → P1 .
Then we interpret the composition T (g) as a new g. Under this interpretation, the union {Qj } of thevalues {g(P1 ), . . . , g(PN )} of g (here g means T (g)) at the punctured points and the values of g at places
where g ramiﬁes are located near 0, the absorbing ﬁxed point of T .
Of course the whole Weierstrass data (g, ω) should be replaced by this composition, i.e., we should
think that the original Weierstrass data should be replaced by the composed pair(eg, ωe ) := (T (g), T∗ (ω)) .
Here, the 1-form part ω of the Weierstrass data is regarded as the holomorphic map Tz M → Tg(z) P1
and T∗ means to take the image by the induced map T∗ acting on tangent vectors of P1 . Then p(eg)
means the operator acting on tangent vectors in TT (g) P1 as the multiplication by p(eg ).
Remark 2.4.1 (gauge invariance of the period condition). The eﬀect of this replacement to theperiod condition should be examined. Let p(g) represent any linear combination of 1 − g 2 , i(1 + g 2 )and 2g (in the Enneper-Weierstrass representation) with R-coeﬃcients a, b, c with a2 + b2 + c2 = 1.
The integrand p(g)ω is understood as follows. Let z be any local coordinate of M and w any aﬃnecoordinate of P1 . Let γ(t) (t ∈ [0, 1]) be a parameterization of γ ∈ H1 (M, Z). Then we haveZ

Z 1p(g)ω =γdw(p(g)ω(γ̇(t)))dt .
0Here the expression of ω depends on the choice of local coordinate functions z in M and w in P1 . This
convention understood, we haveZZp(g)ω =p(eg )e
ω , ∀γ ∈ H1 (M, Z) .
γγ

The period condition implies

Z<p(g)ω = 0 ,∀γ ∈ H1 (M, Z) .

γ

Z

Therefore we have0=<Zp(g)ω = <

γ

p(e
g )e
ω,

∀γ ∈ H1 (M, Z) .

γ

in general. We interpret this as the gauge invariance of the period integral. This means that the period
condition is preserved in the operation (g, ω) 7→ (eg, ωe ). We say that T is a strong hyperbolic translation
if a small circle centered at the repelling ﬁxed point ∞ is mapped to a small circle centered at theabsorbing ﬁxed point 0. By the eﬀect of composing a strong T , the g-image of all punctured points
and critical points locate near the absorbing ﬁxed point 0 where the distance to 0 is measured by the
Fubini-Study metric ωFS of P1 .
From here on we work on the new (eg, ωe ) and therefore we mean by (g, ω) as the new (e
g, ω
e) =
(T (g), T∗ ω) obtained from the original (g, ω) by the application of the strong parabolic translation T .
We return to the discussion on Metrized Riemann-Hurwitz Theorem. Let {Qj } be the collection ofvalues of g at the punctured points and critical points, i.e., those points P satisfying dg(P ) = 0. We

8RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)

proceed to the construction of a good cell decomposition of the basic domain M of a pseudo-algebraic
minimal surface. We pick {Rk } (♯{Rk } = ♯{Qj } − 1) near ∞, the repelling ﬁxed point of T . We
then conformally√ deform ωFS to the pillow case metric (i.e., two euclidean discs of radius slightly
smaller than 2 bridged by highly curved equator) which we regard as a modiﬁed Fubini-Study metricmodiﬁedωFS
on P1 . We make a slit consisting of broken geodesics connecting Q1 , R1 , Q2 , . . . , Ql , Rl , Ql+1
(l + 1 = ♯{Qj }) in this order to make a polygon. Then we develop the polygon onto M via g −1(the inverse map of g) to decompose M into cells. We then take its dual decomposition. After thisprocedure is done, we can make a “microscope” which works well in studying the semi-classical analysis
limr→1 Z(r).
Lemma 2.4.2 [KM1, Part I, Theorem 4.2, Lemma 5.1, Corollary 5.2] (Metrized RiemannHurwitz Theorem). (1) There exists a local parameter ζi centered at each punctured point Pi ∈{P1 , . . . , Pn(M ) } with the following properties :(1-i) The globally deﬁned Poincaré metric ωhyp on M is locally expressed as4|dζi |22
|ζi | (log(c−2 |ζi |−2 ))2, c−2 ≥ 2 .

(1-ii) The pillow case metric interpreted as a modiﬁed Fubini-Study metric is ≥ 4|dζi |2 .
(2) The ratio of the Fubini-Study area against the hyperbolic area of the disk |ζi | ≤ r (r2 < 2) isbounded below by the quantity
r2 log(2/r2 ) .
(3) The stationary value of the local area ratio AreaFS /Areahyp on truncated fundamental domain
F ∩ D(r) for Euclidean distorted fundamental domain whose cluster part has Euclidean height ≤ 1 − r
is at most 2e−1 .
The stationary value calculus based on Lemma 3.4.1 (3) implies the following :α+1

Lemma 2.4.3 [KM1, Part I, Lemma 6.1]. The portion of the annular domain 1 − (1 − r) 2
|z| < r (α < 1) covered by the parabolic sequences has area ratio<

AreaFS /Areahyp ≥ 2e−1
asymptotically as r → 1.
2.5. Collective Cohn-Vossen Inequality.
To interpret Lemma 2.4.3 in terms of the Nevanlinna Theory, we introduce the invariant κg in thefollowing way.
Deﬁnition 2.5.1 [KM1, Part I, Deﬁnition 2.2.4]. Let (M, (g, ω)) be a pseudo-algebraic minimal
surface. The invariant κg is deﬁned asZ rκg = inf κ lim

r→1
exp(κ Tg (t))dt = ∞ .

0

If we deﬁne κhyp similarly by using Thyp (r), we have κhyp (r) = 2. Clearly we have κg > 0. The
invariant κg is characterized by the asymptotic property
κg Tg (r) = log

1
1−rin the sense thatκg = lim

r→1

holds.

as r → 11
log 1−r

Tg (r)

COLLECTIVE COHN-VOSSEN PROBLEM

9Theorem 2.5.2 (Collective Cohn-Vossen Inequality) [KM1, Part I, Theorem 7.1]. For every
pseudo-algebraic minimal surface in R3 , we haveκg ≤ e .
Equivalently, this means
Tg (r) ≥ 2e−1 Thyp (r)⇔Tg (r) ≥ e−1 log

1
.
1−r

2.6. Eﬀective LLD (Lemma on Logarithmic Derivative).
Our approach is based on the eﬀective version of Lemma on Logarithmic Derivative which holds formeromorphic functions on D with small growth. For instance, Theorem 2.5.2 (Collective Cohn-Vossen
inequality) implies that the lifted Gauss map g : D → P1 has small growth.
1∈ [0, ∞) is one to one, weUsing the fact that the correspondence [0, 1) 3 r 7→ ρ(r) := log 1−r
introduce the following symbol : The symbol ≤δ means that the inequality with this symbol holds
outside of a Borel set Eδ in (ρ ∈) [0, ∞) of ﬁnite Lebesgue measure, where the Lebesgue measure of[0, ∞) is understood as dρ.
Let f : D → P1 be a meromorphic function. We deﬁne the invariant κf by




Z r
exp(κTf (t))dt = ∞κf = inf κ lim

r→1

.

0

∂
∂
We interpret df = f (1) as a holomorphic map df : T ′ D → T ′ P1 deﬁned by (df )z ( ∂z
) = f ′ (z) ∂w
where w
1
1
(1)
is the standard aﬃne coordinate of P . For a divisor D on P we deﬁne its ﬁrst jet space as D ⊂ T ′ P1
which consists of points (z, 0) ∈ Tz′ P1 where z ∈ D. We can deﬁne the proximity function mf (1) ,D(1) (r)in the usual way. Moreover, we denote the divisor at inﬁnity which appears in the projective completion
T ′ P1 of T ′ P1 and deﬁne the proximity function mf (1) ,S∞ (r) in the usual way. Then we have

Theorem 2.6.1 [KM2, Part II, Corollary 4.2, Corollary 4.2′ ] (Eﬀective geometric LLD). For
any meromorphic function f : D → P1 satisfying the condition 0 < κf < ∞ and for any small δ > 0,
we have(mf,D (r) − mf (1) ,D(1) (r) ≤δ αLLD (r)(κf + δ) Tf (r)mf (1) ,S∞ (r) ≤δ βLLD (r)(κf + δ) Tf (r)where αLLD (r) + βLLD (r) = 1.
The deﬁnition of κf implies that the RHS is written in terms of κg and Tg (r) : mf,D (r) − mf (1) ,D(1) (r) ≤δ αLLD (r)(1 + δ) log m (1)f,S∞ (r) ≤δ βLLD (r)(1 + δ) log

1= αLLD (r)(κg + δ) Tg (r) ,1−r1= βLLD (r)(κg + δ) Tg (r) .
1−r

2.7. Period Condition Encoding Pair (eH , D) and Characterization of Algebraic Minimal
Surfaces.
The period condition is described in terms of eH , where eH was introduced in 2.2. Lemma 2.2.1 saysthat the period condition is equivalent to the invariance of |eH | under the action of π1 (M )2 . To continue
the study of the period condition in terms of eH , we introduce the D with potentially inﬁnite degree
by setting
D = {eH (z) ∈ P1 | g(z) = ∞} ,2 Note that the original (g, ω) is replaced by the new (eg, ωe ) = (T (g), T∗ ω) and we write it as (g, ω). Therefore it isessential to notice that the period condition is preserved in the application of T (cf. Remark 2.4.1).

10

RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)

i.e., D is a collection of values of eH at places where g has poles. The regularity condition implies thatthe value of eH at any place where g has a pole is well-deﬁned. Restricting the domain of z in D(r)(r < 1), we get a usual divisor of ﬁnite degree. As r approaches to 1, the degree becomes indeﬁnitely
large. So we say that D has potentially inﬁnite degree. We introduce the pair(eH , D)and call it the period condition encoding pair. The main idea of this paper is to apply Theorem
2.6.1 (Eﬀective geometric LLD for meromorphic functions on D with small growth) to (eH , D). In orderto do so, we must show that eH has small growth.
Theorem 2.7.1 [KM2, Part II, Theorem 5.3.3] (growth of TeH (r) under the period condition). Let (M, (g, ω)) be an algebraic minimal surface, i.e., the Weierstrass data satisﬁes the period
coindition. Then, the growth of the order function TeH (r) is comparable to that of the height transform
Thyp (r) of the hyperbolic area function. It therefore follows that 0 < κeH < ∞, i.e.,
∃C>0

s.t.

C log

1
1
< TeH (r) < C −1 log
.
1−r
1−r

Therefore Theorem 2.6.1 applies to the holomorphic function eH : D → P1 with approximation target
D, i.e., eﬀective geometric LLD applies to the period condition encoding pair (eH , D).
The idea behind the proof of Theorem 2.7.1 is the counting rule of measure theoretic quantities
associated to D(r) and therefore essentially relies on Lemmas 2.3.2 and 2.3.3 :Step 1. We decompose
TeH (r) =

1
2

Z 2π
log(1 + |eH |2 )
0

dθ
2π

into contribution from parabolic sequences.
Step 2. We sort the minimum set of fundamental domains covering D(r) into parabolic sequences
starting at fundamental domain with euclidean distortion nth .
Step 3. We integrate the result of Step 1 from, say, 21 to 1 − (1 − r)α(r) against the measuredt
2 log nthdµ(t) =
, where α(r) = 1 −1 .
1−tlog 1−r
Theorem 2.7.2 [KM2, Part II, Lemma 5.3.4, Lemma 5.3.5, Corollary 5.3.6] (characterization of the period condition in terms of the growth of TeH (r)). (1) For any pseudo-algebraic
minimal surface we have

2 
1
TeH (r) = O log.
1−r
Suppose that a pseudo-algebraic minimal surface does not satisfy the period condition. Then TeH (r)21
grows like log.
1−r(2) A pseudo-algebraic minimal surface is algebraic, i.e., the period condition holds, if and only if
1
TeH (r) grows like log
when r → 1, i.e.,
1−r
∃C>0

s.t.

C log

1
1
< TeH (r) < C −1 log
.
1−r
1−r

2.8. Nevanlinna Theory Interpretation of Parabolic Localization Principle.
We want to decode the period condition encoding pair (eH , D) by applying Theorem 2.6.1 (eﬀective
geometric LLD) to extract geometric information from (eH , D). In the decoding procedure, we need toknow how parabolic localization principle is interpreted in Nevanlinna Theory.

COLLECTIVE COHN-VOSSEN PROBLEM

11

2.8.1. Parabolic Localization and Singular Coordinate Change ζ 7→ z.
Let (M, (g, ω)) be a pseudo-algebraic minimal surface. Let z be the standard linear coordinate on D
which is the restriction of the linear coordinate z of C. As we are working on a ﬁnite Riemann surface,
we have another natural coordinate deﬁned by a local parameter ζ around a punctured point P ∈ M \M .
Taking derivatives w.r.to z and ζ, we have two proximity functions mg(1) ,S∞ (r) and mζg(1) ,S∞ (r), wheremζg(1) ,S∞ (r) means the proximity function of g (1) to S∞ taking values in the projective completion T ′ P1
where the derivative is taken w.r.to the local parameter ζ.

Lemma 2.8.1.1 [KM2, Part II, Lemma 5.2.1] (LLD type formula arising from singularcoordinate change ζ 7→ z). The eﬀect of the singular coordinate transform from the local parameterζ at P ∈ M \ M to the lineal coordinate z on D is Nevanlinna theoretically described as
mg(1) ,S∞ (r) − mζg(1) ,S∞ (r) = log

11−r

asymptotically as r → 1 .

2.8.2. Parabolic Localization and Completeness of Induced Metric.
Let (M, (g, ω)) be a pseudo-algebraic minimal surface. The completeness of the metric ds2 on M
f → R3 isinduced from the Euclidean metric of R3 by the Enneper-Weierstrass representation X : M
translated into a Nevanlinna theoretic statement. Let S be the unit sphere in the 3-dimensional R-vector
space V with ON basis { 12 (1 − g 2 ), 2i (1 + g 2 ), g} (cf. 2,2). Then, we have the following interpretationof the parabolic localization in terms of the completeness of ds2 , i.e., the estimate of the magnitude ofmh,S∞ (r) :Lemma 2.8.2.1 [KM2, Part II, Lemma 5.2.2] (Nevanlinna theory interpretation of theparabolic localization in terms of the completeness of ds2 ). Let p(g) represent a generic elementof S. Then we have
TH ′ ,S∞ (r) = 2Tg (r) + 2Ng,∞ (r)andmh,S∞ (r) = 2Tg (r) + 2Ng,∞ (r) .

The proof relies on Lemmas 2.3.2-3 [KM1, Part I, Lemmas 3.4.3.7-8].
2.8.3. Parabolic Localization and Group Theoretic Approximation.
Let (M, (g, ω)) be a pseudo-algebraic minimal surface.
Theorem 2.8.3.1 [KN, Part II, Theorem 5.2.8] (Nevanlinna Theory interpretation of parabolic localization in terms of group theoretic approximation). As soon as we ﬁx the threshold
nth of the strength of Euclidean distortion, we have
(1) Let A(r) ∼ B(r) mean that A(r) and B(r) are the same order as r → 1. Then we have
Ng,∞ (r) ∼ O(n−1th ) log

1
1−r

and

mg,∞ (r) ∼ (1 − O(n−1th )) log1.
1−r(2) The contribution to the hyperbolic area of the minimal covering of D(r) from the intervals J ∈ Jr
over all Jr ’s is bounded above by(1 − r)−1 n−1th .
Here we have deﬁned the collection of intervals Jr in ∂D(r) in [KM2] :Deﬁnition 2.8.3.2 [KM2, Part II, Deﬁnition 5.2.3-4]. (1) For each ﬁxed r satisfying 0 < r < 1 andclose to 1, we deﬁne the subsequence Sr of a parabolic sequence in the following way. The subsequence
Sr starts at the ﬁrst fundamental domain in the parabolic sequence under question whose strength of
Euclidean distortion ﬁrst exceeds nth and ends at the ﬁrst fundamental domain the cluster part of which
has Euclidean height less than 1 − r.

12

RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)(2) Let 0 < r < 1. We decompose ∂D(r) into intervals whose end points are given by those of thecluster part. Let Jr be the collection of intervals corresponding to the subsequence Sr and Ir the rest
(i.e., those intervals corresponding to the cluster part of the parabolic sequence under question whose
cluster part has Euclidean height < 1 − r).
Geometry behind Theorem 2.8.3.1. The geometric origin of Theorem 2.8.3.1 is the group theoretic approximation, which is one of the Nevanlinna theoretic interpretation of the parabolic localization principle. To describe this phenomenon, we recall the structure of the First Main Theorem.
The counting function Ng,w (r) does not depend on the metric of OP1 (1). However, the proximity function mg,w (r) and the order function Tf (r) do depend on the metric. How the asymptotic behavior ofTg (r) as r → 1 depends on the metric is described as follows. If we replace ωFS by another metric
R 2ππωFS + ddc f , then we have Tg,(1/4π)(ωFS +ddc f ) (r) − Tg,(1/4π)ωFS (r) = (1/4π) 0 f (reiθ ) 2π
. In the First
Main Theorem Tg (r) = mg,w (r) + Ng,w (r) − mg,w (0), the dependency on the metric cancels. In our
setting, we operate a strong hyperbolic translation T with ∞ (resp. 0) the repelling ﬁxed point (resp.
absorbing ﬁxed point) so that the g-image of the punctured points and critical points of g is containedin a small disk of 0. By the eﬀect of the application of T , the behavior of Ng,∞ (r) becomes special,while Tg,(1/4π)ωFS(r) by O(1). This observation is important, because
modified (r) diﬀers from Tg,(1/4π)ω
FS
1
Ng,∞ (r) does not depend on the metric of P involved in the deﬁnition of Tg (r). Instead of changing
the metric, we choose to work with a strong hyperbolic translation T in order to make the behavior of
Ng,∞ (r) special (the change of Tg (r) being only by the quantity O(1))3 . In the classical Nevanlinna
theory we have
Mw∈P1 mf,w (r) = O(1)for any meromorphic function f : C → P1 . In our setting, the eﬀect of the Metrized Riemann-Hurwitz
Theorem (Lemma 2.4.2) should be taken into axxount. We recall that in the proof of MRH in [PartI, Theorem 4.2], the Fubini-Study form was deformed by applying a “strong” hyperbolic translation
modiﬁed. In this procedure, the eﬀect that ∞ is
T and further deformed into a pillow-case metric ωFS
the repelling ﬁxed point of T plays an important role. We are ready to describe the phenomenon of
the group theoretic approximation. We ﬁrst subdivide the set-throretical image f (D(t)) ⊂ P1 into theunion of small cells so that any two cells does not share interior points. Then we collect the pre-image
1 ∗ modiﬁedg ωFStaking the multiplicitiesof these cells in D(t) and integrate the area w.r.to the measure 4π
into account and then integrate against dt/t from 0 to r. By this procedure, we get the order function
1
Tg,(1/4π)ωFS
modified (r). The counting function does not depend on the metric of P and the same integration
1
modiﬁedprocedure is performed against the delta measure g ∗ ∞ instead of 4π ωFS. By the eﬀect that ∞is the repelling ﬁxed point in the argument of Metrized Riemann-Hurwitz Theorem [Part I, Theorem4.2], ∃ε > 0 s.th. any open set containing ∞ containing a ball of radius ≥ ε is expanded by g into an
modiﬁedmodiﬁedopen set of S 2 with ωFS
whose boundary is contained in a small disk, say, of ωFS-radius 1/100,
centered at 0 (the absorbing ﬁxed point of T ). Therefore, the cells touching ∂D(r) is counted with highweight in the integration deﬁning the order function Tf,(1/4π)ωFS
modified (r). The weight becomes “heavier”
as T becomes ”stronger”. On the other hand, in the integration deﬁning the counting function Ng,∞ (r),those cells which are counted in the order function with heavy weight are not counted because thosecells locate near the boundary ∂D(r) because of the eﬀect of the parabolic localization (see below). By
the eﬀect of the parabolic localization, even under the limit of subdivisions so thatmax{diam ωFS (cells)} → 0, the ratio of the number (counted with multiplicities) of cells whose pre-image
by g intersects the boundary of D(r) against that of all cells in the subdivision indeﬁnitely increases as
r → 1. We note that(1) the multiplicities of those cells whose pre-image under g intersects ∂D(r) diverges to ∞ with1
as r → 1,
order 1−r
and that(2) the signiﬁcant portion of the collection of those cells in P1 whose radius (w.r.to ωFS ) is of order1 − r and whose pre-image under g intersects the boundary ∂D(r) is not counted in the counting function, while they are contained in the order function.
3 This is the gauge transformation induced by the application of T by which the period condition is preserved, i.e.,
the new pair (T (g), T∗ ω) satisﬁes the period condition just as (g, ω) does.

COLLECTIVE COHN-VOSSEN PROBLEM

13

The above observation implies that the ratio of Ng,∞ (r) against Tg (r) (in fact Tg,(1/4π)ωFS
modified (r))
becomes signiﬁcantly small when r approaches to 1. We call this phenomenon as the group theoretic
approximation introduced by the parabolic localization and Metrized Riemann-Hurwitz Theorem(Lemma 2.4.2). This approximation is NOT an approximation for the original holomorphic map g to
∞ but is a RELATIVE approximation which means that Ng,∞ (r) occupies only a small portion of Tg (r)modiﬁedw.r. to the pillow-case metric (1/4π)ωFS
of P1 arising from the application of the strong hyperbolictranslation T . This phenomenon does not happen for entire holomorphic map f : C → P1 . Indeed, inthe limit of subdivisions so that max{diam ωFS (cells)} → 0, the ratio of the number of cells intersecting∂D(r) against the that of all cells in subdivisions tends to 0 (counted with multiplicities).
Proof of Theorem 2.8.3.1. Before the proof, we recall that by rotation (if necessary), we may assume
that the extended Gauss map g does not have poles at punctured points and in addition that g has
only simple poles. We recall that in this setting the point ∞ is the repelling ﬁxed point of a strong
hyperbolic translation T used in Metrized Riemann-Hurwitz Theorem [Part I, Theorem 4.2]. The proof
consists of the justiﬁcation of the intuitive explanation in the above speculation based on the meaningof ∞, i.e., the strong hyperbolic translation T in the Riemann-Hurwitz Theorem has ∞ as the repellingﬁxed point. Then Ng,∞ (r) has a canonical meaning. More generally, so does Ng,a (r), where a is any
value which does not belongs to the set {g(P ) | P ∈ M \ M }) as soon as a is chosen as the repellingﬁxed point of the strong hyperbolic translation in the proof of Metrized Riemann-Hurwitz Theorem.
To explain the reason why Ng,∞ (r) has a canonical meaning, we work on H. Consider a sequenceof fundamental domains having ∞ as a parabolic ﬁxed point. Then the parabolic sequence consistsof parallel translations of a reference fundamental domain. Suppose that the Euclidean height of thecluster part is 1. The boundary ∂D(r) corresponds to the circle
Cr : r =|z − i|.
|z + i|Put ε = 1 − r. Then the radius of Cr is of order ε−1 . The lower arc of the circle Cr is approximated by
the parabola of the formy = ε2 + εx2 .
The number of fundamental domains in the parabolic sequence which have non-empty intersection with1
Cr is of order O(ε− 2 ). Therefore, the hyperbolic area of the portion of the parabolic sequence truncated1by D(r) having non-empty intersection with ∂D(r) is at most of order O(ε− 2 ). On the other hand, thehyperbolic area of the rest of the parabolic sequence truncated by D(r) is
Z
O

1
εN

Z ε−1

dy
dx
2
−1
ε 2
ε2 +εx2 y



Z
=O

1
εN

dx
− 1 ε2 + εx2ε 2= O(ε−2 ) ,where N is a large number but  ε− 2 which appears when we approximate Cr by the parabolay = ε2 + εx2 . This manipulation implies that the contribution to the hyperbolic area from the parabolic
sequence is dominated by those fundamental domains whose Euclidean height is smaller than 1 − r.
To sum up, the counting function Ng,∞ (r) (more generally, Ng,a (r), where −1/a, a 6∈ {g(P ) | P ∈M \ M or dg(P ) = 0}) behaves in a special way, as soon as the original (g, ω) is replaced by a new (eg, ωe)by the application of the strong hyperbolic translation T having {a} (resp. {−1/a}) as the repelling(resp. absorbing) ﬁxed point. The above manipulation is the mathematical ground for this statement.
Indeed, as is shown in the following discussion, the growth of Ng,∞ (r) (more generally, Ng,a (r), where1
as soon as we take the eﬀect of the application{a} is as above) is strictly smaller than that of log 1−r
of T into account.
We count how many members of a parabolic sequence contribute to Ng,∞ (r). It follows from [Part I,
Lemma 3.4.3.9-10] that we have only to look at a parabolic sequence starting at a fundamental domainof Euclidean diameter (1 − r)α(r) where1α(r) = 1 −2 log nth,1
log 1−ri.e., nth = (1 − r)α(r)−1
2.

14

RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)

Applying the nth -th iteration of the parabolic translation to the initial fundamental domain the Euα(r)+1
clidean diameter decays like O((1 − r) 2 ) and the Euclidean diameter of the cluster part decays
like O(1 − r). Therefore, it follows from the above manipulation that the number of fundamental doα(r)−1mains which contribute to Ng,∞ (r) is just O((1 − r) 2 ). We have approximately O((1 − r)−α(r) )such fundamental domains and the contribution from such ones dominates by [Part I, Lemma 3.4.3.10].
Therefore, the contribution to Ng,∞ (r) is just of order
Z r

α−1
dt
(1 − t)−α (1 − t) 2 ≈
2−1 t

Z r

1−α
dt
−1
(1 − r)−1 (1 − t) 2 ≈ n−1.
th log(1 − r)2−1 t(2) We measure the area of the portion of D(r) occupied by fundamental domains corresponding
to |Jr | := ∪J∈Jr J. The contribution to the area from each |Jr | is bounded above by the number of
α(r)−1intervals in Jr . This is of order nth = (1 − r) 2 . Therefore the contribution to the area of theminimal covering of D(r) from |Jr | is bounded above by(1 − r)−α(r) nth = (1 − r)−
We have thus proved Theorem 2.8.3.1.

α(r)+1
2= (1 − r)−1 (1 − r)

1−α(r)
2= (1 − r)−1 n−1
th .

□2.9. From LLD to SMT.
Let (M, (g, ω)) be a pseudo-algebraic minimal surface. We interpret dg = g (1) dz and ω = h(z)dz =′′h(ζ)dζ as a map from T ′ D to T ′ P1 , i.e., dg = g (1) : Tz′ P1 → Tg(z)P1 and h : Tz′ D → Tg(z)P1 (∀z ∈ D).
Following the classical argument from LLD (Lemma on Logarithmic Derivative) to SMT (the Second
Main Theorem) (cf. [N] and [Kod]), we have the following Nevanlinna calculus inequality.
Lemma 2.9.1 [KM2, Part II, Lemma 5.1.1] (Nevanlinna calculus inequality for the lifted
Gauss map). Let (M, (g, ω)) be a pseudo-algebraic minimal surface and (g, ω). Let D be a ﬁnite
collection of distinct points which is regarded as a divisor on P1 . Then the Nevanlinna theory function
mg,D (r) + Ng,Ram (r) which counts the number of solutions of the equation g(z) ∈ D (counted withmultiplicity) satisﬁes the following estimate
mg,D + Ng,Ram (r) ≤δ (mh,S0 (r) + Nh,S0 (r) − mh,S∞ (r)) +(κg + δ) Tg (r) − J(r){z}|
Th,S0 −S∞ (r)=2Tg (r)asymptotically as r → 1, where J(r) is the Nevanlinna theoretic function deﬁned byJ(r) := mg(1) ,S∞ (r) − mζg(1) ,S∞ (r) + mh,S0 (r) − mζh,S0 (r) .
Therefore, the RHS of Lemma 2.9.1 becomes (omitting δ)
2 Tg (r) + {κg Tg (r) − J(r)} .
It follows from the proof in [KM2, Part II, Lemma 5.1.1] that the origin of 2 before Tg (r) is topological,
i.e., 2 = χ(P1 ), the Euler number of the 2-sphere. This is the same as the classical Second Main
Theorem. On the other hand, the origin of the quantity κg Tg (r) − J(r) is Theorem 2.6.1 (the Eﬀective
geometric LLD) and has group theoretic and analytic in nature.
It therefore turns out that evaluating the Nevanlinna theoretic function κg Tg (r) − J(r) is the maintask toward establishing the Second Main Theorem for the lifted Gauss map g : D → P1 .
2.10. Proof of the Second Main Theorem for the Lifted Gauss Map of an Algebraic
Minimal Surface.
In 2.10, we decode the period condition encoding pair (eH , D) by applying Theorem 2.6.1 (theEﬀective geometric LLD). We have two versions of the eﬀective LLD applied to (eH , D). One is “without
ζ”, i.e., the derivative is taken w.r.to the linear coordinate z on D ⊂ C. The other is “with ζ”, i.e., thederivative is taken w.r.to the local parameter ζ around a punctured point P ∈ M \M = {P1 , . . . , Pn(M ) }.

COLLECTIVE COHN-VOSSEN PROBLEM

15

In order to understand the “hidden” geometry of algebraic minimal surface which should be extracted
from the two types of eﬀective LLD’s applied to (eH , D), we introduce the following Nevanlinna theoretic
functions(2.10.1)

J(r) := (mg(1) ,S∞ (r) − mζg(1) ,S∞ (r)) + (mh,S0 (r) − mζh,S0 (r)) ,J1 (r) := mh,S0 (r) − mζh,S0 (r) ,J2 (r) := mh,S∞ (r) − mζh,S∞ (r)which are constructed from the singular coordinate transformation from the local parameter ζ at punctures {P1 , . . . , Pn(M ) } to the linear coordinate z of D ⊂ C. These are Nevanlinna theoretic functions
which essentially depend only on the conformal structure of the basic domain M .
For instance, Lemma 2.8.1 implies thatmg(1) ,S∞ (r) − mζg(1) ,S∞ (r) ≥ log

11−r

holds. In terms of J(r) and J1 (r), this formula is rewritten as(2.10.2)

J(r) ≥ J1 (r) + log

1
.
1−r

In the following sequence of Lemmas 2.10.1-4, we compare two types of eﬀective LLD applied to thepair (eH , D) and extract useful information from the period condition. The inequalities which we will
prove in Lemmas 2.101-4 involve small δ > 0 and the associated occurrence of the exceptional intervals
in the same sense as that explained at the beginning of 2.6. For simplicity we use the symbol ≤δ as in
2.6.
Let us write
Ar := {a1 , . . . , ak(r) }for the totality of the values of eH at the poles of g inside |z| = r (0 < r < 1) where z is the linear
coordinate of D ⊂ C. Here, for a ﬁxed r < 1, the number k(r) which appears in the deﬁnition of Ar
is ﬁnite and limr→1 k(r) = ∞. The image curve g(∂D(r)) in M must be at distance from the set of
all punctured points M \ M by some positive number. As |eH | is invariant under the action of π1 (M ),these values of eH are expressed as(2.10.3){α1 eiθ11 , . . . , α1 eiθ1k1 (r) , . . . , αp eiθp1 , . . . , αp eiθpkp (r) }|{z}ﬁnitely many variations for |ai |=α1 ,...,αp even if k(r)→∞where k(r) = k1 (r) + · · · + kp (r) and p ≤ ♯(g −1 (∞)) (equality, if counted with multiplicities) whereg : M → P1 is the Gauss map. The picture how these points (2.10.3) are distributed on the Riemannsphere P1 is as follows. There exist a ﬁnitely many circles(2.10.4)|w|P1 = αi

( i = 1, . . . , p ) inP1

and the points (2.10.3) distribute on these circles. Here the parameter w in the expression |w|P1 = αi is
a ﬁxed aﬃne coordinate of P1 . What is essential is that the point {∞} ∈ P1 is not an accumulation pointof the point set (2.10.3). Indeed, as p(g) varies in the unit sphere S(V ) in V , the location of the circles|w|P1 = αi in P1 is uniform and is uniformly apart from the point {∞}. As k(r) → ∞ as r → 1, the pointsin (2.10.3) distribute more densely on ﬁnitely many circles (2.10.4), i.e., ∪i {|w|P1 = αi } (1 ≤ i ≤ p)
in P1 and never accumulate at ∞. This means that although the “divisor” D has potentially inﬁnite
degree “as r → 1”, we can imagine that the set consisting of the ﬁxed circles |w|P1 = αi (1 ≤ i ≤ p)
behaves as a usual “ﬁxed target divisor” on P1 when we discuss the approximation of eH to its values
at the poles of the lifted Gauss map g.

16

RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)

Lemma 2.10.1 (Eﬀective LLD “without ζ” applied to (eH , D) for algebraic minimal surfaces)(cf. [KM2, Part II, Lemma 5.4.1]). Let (eH , D) be the period condition encoding pair of any
algebraic minimal surface. Let the notations such as ∪i {|w|P1 = αi } be understood as in (2.10.4). ThePk(r)eﬀective LLD without the superscript ζ for the function eH with the divisor i=1 {ai } is the following
inequality (δ being any small positive number).
Mp(g)∈S {(meH ,∪i {|w|P1 =αi } (r) − m(eH )(1) ,∪i ∪|w|
≤δ (1 + δ)κg Tg (r)1= (1 + δ) log
1−r

asymptotically as=αiP1{z}(1) (r)) + m(eH )(1) ,S∞ (r)}r→1.

Lemma 2.10.2 (Eﬀective LLD “with ζ” applied to (eH , D) for algebraic minimal surfaces)(cf. [KM2, Part II, Lemma 5.4.2]). Let (eH , D) be the period condition encoding pair of any
algebraic minimal surface. Let the notations such as ∪i {|w|P1 = αi } be understood as in (2.10.4). The
eﬀective LLD with the superscript ζ for the function eH with the divisor∪i {|w|P1 = αi } + {0} + {∞}is the following inequality (δ being any small positive number).
Mp(g)∈S {(mζeH ,∪i {|w| 1 =αi } (r) − mζ(eH )(1) ,∪i ∪P|w| 1 =αi {z}
P

(1)

(r))

+ (mζeH ,{0} (r) − mζ(eH )(1) ,{0}(1) (r)) + (mζeH ,{∞} (r) − mζ(eH )(1) ,{∞}(1) (r))+ mζ(eH )(1) ,S∞ (r)}≤δ (1 + δ) κg Tg (r)1= (1 + δ) log
1−r

asymptotically as

r→1.

The proofs of Lemmas 2.10.1-2 are almost the same as those of [KM2, Part II, Lemmas 5.4.1-2].
We have only to replace the LLD-type formulas in [KM, Part II, (5.2.9)], which was proved under theassumption that the Cohn-Vossen ratio R ≤ 2, with the usual one, i.e., Theorem 2.6.1 (the eﬀective
geometric LLD) of type
mg,D (r) − mg(1) ,D(1) (r) ≤ αLLD κg Tg (r) ,mg(1) ,S∞ (r) ≤ αLLD κg Tg (r) .
This is the point which explains the reason why the case R = 2 is so special.
A small technical point is that the interval version of κg appears if we try to adapt the argument
in [KM2] to the proof of Lemma 2.10.1. For instance, if the interval version appears, the deﬁnition
κg (I)κgκb (I) := k(r)should be replaced by the deﬁnition κb := k(r)and the relation (κb (I)1(I)) · Tgb (r) =|I|
11
2π log 1−r should be replaced by the relation κb Tgb (r) = log 1−r .

Another diﬃculty is that k(r) diverges as r → 1. By using Lemma 2.3.4 [KM1, Part I, Lemmas
3.4.3.9-10], we settle the diﬃculty arising from the fact k(r) → ∞ (as r → 1) in Lemmas 2.10.1-2.
Our next task is to translate the estimates in Lemmas 2.10.1-2 into Nevanlinna theory statements
by using the Nevanlinna theory interpretations of the parabolic localization principle explained in 2.8.
We have the following results.
Lemma 2.10.3 (Consequence of eﬀective LLD “without ζ” applied to (eH , D)) (cf. [KM2,
Part II, Lemma 5.4.3]). Let (eH , D) be the period condition encoding pair of any algebraic minimal
surface. The eﬀective LLD in Lemma 2.10.1 without the superscript ζ implies the following inequality(δ being any small positive number).
J2 (r) + mg,∞ (r) ≤δ (1 + δ) κg Tg (r) = (1 + δ) log

11−r

asymptotically as

r→1.

COLLECTIVE COHN-VOSSEN PROBLEM

17This implies
J2 (r) ≤δ (1 + δ) κg Tg (r) − Tg (r) + Ng,∞ (r)asymptotically as

r→1.

As soon as we replace [KM2, Part II, Lemma 5.4.1] with Lemma 2.10.1, we can prove Lemma 2.10.3
exactly in the same way as [KM2, Part II, Lemma 5.4.3].
Lemma 2.10.4 [KM, Lart II, Lemma 5.4.4] (Consequence of eﬀective LLD with ζ appliedto (eH , D)). Let (eH , D) be the period condition encoding pair of any algebraic minimal surface. The
eﬀective LLD in Lemma 2.10.2 with the superscript ζ implies the following inequality (δ being any small
positive number) :(8κ−1
g − 2 + δ) · log1≤δ J1 (r) + 2 J2 (r) + 2Ng,∞ (r)1−rasymptotically as

r→1.

Proof. The proof reduces to showing the following inequalities :(2.10.5a)M {mζeH ,∪i |w| 1 =αi (r) − mζ(eH )(1) ,∪i ∪P|w| 1 =αi {z}P(1)(r)} ≥1 ζ
m ′ (r) + 2mg,∞ (r) ,2 H ,S0(2.10.5b)(mζeH ,{0} (r) − mζ(eH )(1) ,{0}(1) (r)) + (mζeH ,{∞} (r) − mζ(eH )(1) ,{∞}(1) (r)) = mζh,S∞ (r) ,(2.10.5c)mζ(eH )(1) ,S∞ (r) = O(1) .
In order to prove (2.10.5a) it suﬃces to proveXk(r){mζeH ,{ai } (r) − mζ(eH )(1) ,{ai }(1) (r)} ≥

i=1

1 ζ
m ′ (r) + 2mg,∞ (r)2 H ,S0

for generic p(g) ∈ S(V ). To prove Lemma 2.10.4, we ﬁrst assume that the estimates (2.10.5a.b.c) hold.
Then the consequence of Lemma 2.10.2 becomes(2.10.6)1 ζ
m ′ + 2mg,∞ + mζh,S∞ (r) ≤δ (1 + δ) κg Tg (r) .
2 H ,S0

Note that here we do not need to take the mean over S(V ), because what contributes to the LHS of(2.10.6) is only the set of points of D where g has poles.
Whenever (eﬀective) LLD is used there occurs a small constant δ and the inequality ≤ becomes ≤δ .
With this understood, for simplicity, we simplify formulas by omitting small constant δ and replacing≤δ and 1 + δ by ≤ and 1.
The deﬁnition of J1 (r) impliesmζH ′ ,S0 (r) = mH ′ ,S0 (r) − J1 (r) .
Lemma 2.8.2.1 (parabolic localization in terms of the completeness of the Riemannian metric of M
induced from R3 ) imples
TH ′ ,S∞ (r) = 2 Tg (r) + 2 Ng,∞ (r) .
Using these two estimates, we havemH ′ ,S0 (r) = TH ′ ,S0 (r) [ NH ′ ,S0 (r) = 0 ]= TH ′ ,S∞ +OP1 (2) (r)[ linear equivalence [S0 ] = [S∞ ] + OP1 (2) ]= 2 TH ′ ,S∞ (r) + 2 Tg (r)= 2 Tg (r) + 2 Ng,∞ (r) + 2 Tg (r) [ Lemma 2.8.2.1 ]1= 4 κ−1+ 2Ng,∞ (r) .
g log
1−r

18

RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)

On the other hand, we havemh,S∞ (r) = Th,S∞ (r)= Th,S0 −OP1 (2) (r) [ linear equivalence [S0 ] = [S∞ ] + OP1 (2) ]= Th,S0 (r) − 2 Tg (r)= mh,S0 (r) + Nh,S0 (r) − 2 Tg (r) [ First Main Theorem ]= 2 mg,∞ (r) + 2 Ng,∞ (r) − 2 Tg (r) [ Regularity condition ]= O(1) .
As
J2 (r) = mh,S∞ (r) − mζh,S∞ (r) ,we havemζh,S∞ (r) = mh,S∞ (r) − J2 (r) = −J2 (r) + O(1) .
Inserting these into (2.10.6) we have the consequence of Lemma 2.10.4. Indeed, we have
1 ζm ′(r) + 2 mg,∞ − J2 (r)2 H ,S∞1
1≥ mH ′ ,S0 (r) − J1 (r) + 2 mg,∞ (r) − J2 (r)2211= (4 Tg (r) + 2 Ng,∞ (r)) − J1 (r) + 2 mg,∞ (r) − J2 (r) .
22

κg Tg (r) ≥It follows that
J1 (r) + 2J2 (r) ≥ −2 (κg Tg (r) + 4 Tg (r) + 2 Ng,∞ (r) + 4 mg,∞ (r)= −2κg Tg (r) + 6 Tg (r) + 2 mg,∞ (r) .
Adding 2Ng,∞ (r) to both sides, we have
J1 (r) + 2J2 (r) + 2 Ng,∞ (r) ≥ −2κg Tg (r) + 8 Tg (r)1≥ (8κ−1.
g − 2) · log
1−r
The rest of the proof consists of showing (2.10.5abc) and is the same as that of [KM2, Part II, Lemma
5.4.4]. The hardest is the proof of (2.10.5a). The eﬀect of the group theoretic approximation, whicharises from the application of the strong hyperbolic translation T with ∞ (resp. 0) the repelling (resp.
absorbing) ﬁxed point, should be taken into account. Because (2.10.5a) is highly non-trivial, we repeat
its proof. The eﬀect of the application T is present in the occurrence of 2mg,∞ (r) in (2.10.5a). Supposethat H ′ = p(g)ω = 0. Then, eH stays longer at points where p(g) = 0. Hence the places where p(g) = 0
gives rise to the contribution to the proximity function m(eH )(1) ,S0 (r). If p(g) = 0, then the value of g isa solution of the equation p(w) = 0 (e.g., p(w) = 1 − w2 , p(w) = i(1 + w2 ) and so on), where p ∈ S(V ).
modiﬁed
Therefore the value of g is generically far from 0 w.r.to the pillow case metric ωFS. This means
2that if g solves the equation p(w) = 0 (e.g., if p(w) = 1 − w , then w = g = ±i), then the solutionof the equation p(g) = 0 is generically close to the place where g has a pole (in particular, far fromthe pre-image g −1 ({g(P ) | P ∈ M \ M }). Theorem 2.8.3.1 implies that the eﬀect of the group theoretic
approximation implies
mg,w (r) = (1 − O(n−1th )) · Tg (r)for ∀w = g(p) where p is contained in a suﬃciently small neighborhood of the set of all points where
g has a pole. Therefore, p(g) = 0 happens at places in the domain of g which are close to the polesof g in the sense that p(g) = 0 has solutions at places in a small punctured neighborhood of poles ofg where p ∈ S(V ) is generic. Thus, the approximation of p(g) to 0 gives rise to the approximationof g to a small neighborhood of places where g has a pole and therefore, by the regularity condition,
the “extra approximation of eH to D”. Here, the “extra approximation of eH to D” should not beCOLLECTIVE COHN-VOSSEN PROBLEM

19

understood literally. Instead, this should be understood as the approximation of eH to the values of
eH at places where p(g) = 0 which occurs via the approximation of (eH )(1) to S0 . We have shown in[KM2, Part II, Lemma 5.3.3] that the approximation of g to ∞ contributes only to the approximationof eH to D. We can argue in the parallel way that the eﬀect of this “extra approximation to D” canbe estimated in a similar way. Since the equation p(w) = 0 is a quadratic equation, we infer thatthe “extra approximation of eH to D” in question is generically of magnitude 2 mg,∞ (r). Indeed, we
can work separately on two roots of the equation p(w) = 0. In this case, by using the argument in
the proof of [KM2, Part II, (5.4.15) and (5.4.17)], we conclude that “the extra approximation to D”
is mg,w1 (r) + mg,w2 (r) (w1 , w2 being the places where p(g) = 0 occurs) which is generically of themagnitude mg,∞ (r) + mg,∞ (r) = 2mg,∞ (r) by the group theoretic approximation described in Theorem
2.8.3.1. Taking [KM2, Part II, (5.4.15) and (5.4.17)] (i.e., 32 mζH ′ ,S0 (r) − mζH ′ ,S0 (r) = 12 mζH ′ ,S0 (r)) andthe extra approximation stemming from the solution of p(g) = 0 (i.e., 2mg,∞ (r)) into account, we havethe RHS of (2.10.5a). □Lemma 2.10.5 (Bounding κg Tg (r) − J(r)) (cf. [KM2, Part II, Lemma 5.4.5]). Let (M, (g, ω))be an algebraic minimal surface. Then we haveκg Tg (r)−J(r) ≤ (4−10κ−1g ) log1≤ (4κg −10) Tg (r) ≤ (4e+ −10) Tg (r)1−rasymptotically as

r→1.

Proof. For simplicity we omit small δ > 0 which appears in applying LLD. Recall Lemmas 2.10.3-4, i.e.,
J2 (r) ≤ κg Tg (r) − Tg (r) + Ng,∞ (r)and(8κg−1 − 2) · log

1≤ J1 (r) + 2 J2 (r) + 2Ng,∞ (r) .
1−r

It follows that1
− 2 J2 (r) − 2 Ng,∞ (r)1−r
11≥ (8κ−1− (2 − 2κg−1 ) · log− 4 Ng,∞ (r)g − 2) · log
1−r
1−r
1
= (10κ−1− 4 Ng,∞ (r) .
g − 4) · log
1−r

J1 (r) ≥ (8κ−1
g − 2) · log

We have from Lemma 2.8.1.1 (LLD type interpretation of parabolic localization) the estimate
J(r) ≥ J1 (r) + log

1
.
1−r

Therefore, we haveJ(r) ≥ J1 (r) + log

1
1≥ (10κ−1− 4 Ng,∞ (r) .
g − 3) log
1−r
1−r

Therefore11− (10κ−1+ 4 Ng,∞ (r)g − 3) · log
1−r
1−r
1
= (4 − 10κ−1+ 4 Ng,∞ (r)g ) · log
1−r
≤ (4κg − 10) · Tg (r) + 4 Ng,∞ (r) + O(n−1th )Tg (r) .

κg Tg (r) − J(r) ≤ log

Finally, we use the Theorem 2.8.3.1 (group theoretic approximation arising from parabolic localization),
namely, we use the estimate
Ng,∞ (r) ≤ O(n−1th )Tg (r) .
We haveκg Tg (r) − J(r) ≤ (4 e − 10) Tg (r) + O(n−1th )Tg (r) ,which ﬁnishes the proof of Lemma 2.10.5.

□

20

RYOICHI KOBAYASHI (JOINT WITH REIKO MIYAOKA)

Theorem 2.10.6 (Second Main Theorem for Gauss map of algebraic minimal surfaces) (cf.
[KM2, Part II, Theorem 5.4.6]). The lifted Gauss map of an algebraic minimal surface (M, (g, ω))satisﬁes the Second Main Theorem
mg,D (r) + Ng,Ram (r) ≤ {4(e − 2) + O(n−1th )} Tg (r) .
The inequality should be understood that ≤ (resp. 4(e − 2)) means ≤δ (resp. (4(e − 2) + δ)).
Proof of Theorem 2.10.6. By rotating a given algebraic minimal surface in R3 , we may assume without
loss of generality that(i) {∞, 0} is NOT contained in the union of g(M \ M ) and the g-image of the set P s.t. dg(P ) = 0,(ii) {∞, 0} is NOT contained in Supp(D).
In this setting, we may use Theorem 2.8.3.1. Combining Lemma 2.9.1 and Lemma 2.10.6 we have
Theorem 2.10.6. □In particular, Theorem 2.10.6 implies that the totally ramiﬁed value number νg (resp. the number
Dg of the exceptional values of g : M → P1 ) satisﬁes
Dg ≤ νg ≤ 4(e − 2) ≤ 2.88 .
This settles Osserman’s problem, i.e., the Gauss map of an algebraic minimal surface can omit at most
two values.
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Ryoichi Kobayashi
ryoichi@math.nagoya-u.ac.jp
Graduate School of Mathematics, Nagoya University